[This is from a very neat example my real analysis professor used some years ago. While I'm fairly confident it's neat, I'm not certain it's top-level-post-worthy. The general point is about problems with applying concepts involving infinity to reality; any advice on content (or formatting!) would be greatly welcomed. My math education basically ended after a few upper division courses, so it's possible there are some notational schemes or methods I am ignorant of.
I think this is a fun little exercise, if nothing more.]
The concept of "infinity" and "infinite series" and sets get thrown around a lot in mathematics and some of philosophy. It's worth trying to put the concept of infinity in perspective before we try to think of things in the real world being "infinite." Warning: this post will involve numbers that are literally too large to comprehend. But that's the point.
Let us define an operator, /X\ ("triangle-X"). /X\ = X raised to the X power X times. Thus, /2\ = 22^2 = 24 = 16.
//2\\ ("2-triangle-2") would do this operation twice. Thus, it would equal /16\, the value of which we'll get to in a minute.
We now introduce a new operator, [X] ("square-X"). [X] = triangle-X-triangle-X, i.e. X inside of /X\ triangles.  = ////////////////2\\\\\\\\\\\\\\\\
We can introduce another operator, [X> ("pentagon-X"). [X> = X inside of [X] squares. I believe this would be "square-X-square-X").
[Edited for clarity]
I'll spare the next [X] operators, and go right to (X) ("circle-X"). Technically, it's whole-lot-of-sides-polygon-X - we could continue this process indefinitely - but we'll call it circle-X, because that's as far as we're going. (X) follows the process that took us from triangle to square to pentagon, iterated an additional [X] times.
I'll be honest. This got kind of meaningless a bit before [X]. Let's start trying to construct what  equals, and you'll see why.
/2\=16. So //2\\ = /16\ = 1616^16^16^16^16^16^16^16^16^16^16^16^16^16^16^16. Using some very rough approximations, we can say this is about 102x10^19, or one followed by twenty billion billion zeroes. //16\\ is thus one followed by twenty billion billion zeroes, raised to the power of one followed by twenty billion billion zeroes one followed by twenty billion billion zeroes times. My math education could be more complete, but I am not aware of another way to denote such a number. To say it could not be written in scientific notation on a universe-sized sheet of paper is probably a colossal understatement. And after we calculate that number we have to repeat the process thirteen more times to get . We could theoretically keep doing this until we got to (2); (2) is a number that cannot be meaningfully expressed, understood, or calculated by any means that exist today. And there's still ((2)) after that.
Now, imagine that this period (.) represents zero. Imagine drawing a line from that point to one on the near surface of the sun, which represents infinity (yes, this is improper - it's a finite line - but the point is visualization, so understatement really isn't an issue here). (2) lies within the parentheses surrounding that period, and that's an understatement of how close it is to zero. It really isn't even 1/(2) inches from that period.
Remember this the next time you ponder the meaning, use, or existence of an infinite set, infinite repetitions, or an infinite time.